1.4 The Dilemma of Light

According to Galilean relativity, inertial reference frames constitute a set of coordinate systems in terms of which a large class of (idealized) "free" motions are linear, and in terms of which the inertia of resting objects is independent of spatial direction (i.e., isotropic). Also, every material object in any state of motion is momentarily at rest with respect to one of these frames. These coordinate systems proved to be extraordinarily useful for describing and analyzing the motions of material objects. However, some of the motions occurring in nature, notably the propagation of light, seem incompatible with the Galilean framework of inertial coordinate systems. If light is a material substance, then according to Galilean relativity there should be an inertial reference frame with respect to which light is at rest in a vacuum, but in fact we never observe light in a vacuum to be at rest with respect to any inertial reference frame. On the other hand, if light is a wave propagating through a material medium, then the constituent parts of that medium should, according to Galilean relativity, behave inertially, and in particular should have a definite rest frame, whereas we find that light propagates best through a vacuum, a region in which no (detectable) material with a definite rest frame exists. Since light conveys information and energy across intervals of space and time, it is obviously a significant physical phenomenon, but it's behavior defies realistic representation within the framework of Galilean inertial coordinate systems.

However, we should note that the behavior of light is perfectly compatible with the fundamental premise of Galilean relativity, which (as summarized above) is that there exist inertial coordinate systems in terms of which free motions are linear, inertia is isotropic, and every material object is instantaneously at rest with respect to one of these systems. None of this conflicts with the observed behavior of light, because the motion of light is observed to be both linear and isotropic with respect to inertial coordinate systems. The fact that light is not at rest with respect to any system of inertial coordinates does not conflict with Galileo's premise if we agree that light is not a material object.

The incompatibility of light with Galilean relativity arises not from the principle of relativity itself (G1), but from the auxiliary assumption that Galileo (and Newton) tacitly incorporated into the framework, namely, the assumption that two relatively moving coordinate systems are related to each other in such a way that the composition of co-linear speeds is simply additive (G2). As discussed in the previous section, we aren't free to impose this assumption on the class of inertial coordinate systems, because they are fully determined by the requirements of linear free motions and inertial isotropy. There are no more adjustable parameters (aside from insignificant scale factors). The composition of velocities of relatively moving inertial coordinate systems is a matter of empirical fact.

Furthermore, we saw in the previous section that the most physically meaningful definition of the velocity between two entities is their reciprocal velocity with respect to each others' rest frames, whereas a pulse of light in a vacuum has no (non-singular) rest frame, so it isn't clear how to compose a massive object's velocity with the velocity of light - unless the speed of light with respect to arbitrary inertial coordinates is singular. This suggests that in order to include the phenomenon of light (and other motions at nearly the speed of light) into a common framework of space and time with the motions of material bodies, it will be necessary to modify G1 or G2 (or both). We have some freedom of choice in this, to the extent that our coordinate systems are matters of convention. One set of compatible principles, proposed by Einstein in 1905, can be paraphrased as follows

(E1) For any massive body, in any state of motion, there exists a system of space and time coordinates (called "inertial coordinates") with respect to which that body is instantaneously at rest, free objects move uniformly in straight lines, and inertia is the same in all spatial directions.

(E2) If a material object B is moving at the speed v with respect to the inertial rest frame coordinates of a material object A, and if an object C is moving in the same direction at the speed u with respect to the inertial rest frame coordinates of B, then C is moving at the speed (v + u)/(1 + uv/c²) with respect to the inertial rest frame coordinates of A, where c is the speed of light in a vacuum.

Obviously E1 is identical to G1 of the previous section, which is to say, the fundamental principle of relativity adopted by Einstein is the same as Galileo's. Furthermore, E2 would be identical to Galileo's G2 - if the speed of light was infinitely great, which was still considered a real possibility in Galileo's day. Many people, including Descartes, regarded rays of light as instantaneous. Even Newton's Opticks, published in 1704, made allowances for the possibility that "light be propagated in an instant" (although Newton himself was persuaded by Roemer's observations that light has a finite speed). Hence it can be argued that the principles of Galileo and Einstein are essentially identical in both form and content. The only difference is that Galileo assessed the propagation of light to be "if not instantaneous then extraordinarily fast", and thus could neglect the term uv/c², especially since he restricted his considerations to the movements of material objects, whereas subsequently it became clear that the speed of light has a finite value, and it was necessary to take account of the uv/c² term when attempting to incorporating the motions of light and high-speed particles into the framework of mechanics.

We've expressed E2 in a way that emphasizes its formal similarity to G2, but it's easy to see that the same content could be captured by the alternative form

(E2') The speed of light with respect to any system of inertial coordinates is isotropic and independent of the state of motion of the source.

This form emphasizes the fact that this principle can be regarded as an assertion of wave-particle duality for light, because if light behaves like material particles its speed ought to be isotropic with respect to any inertial frame, and if light consists of a wave in a material medium its speed ought to be independent of the state of motion of the source, whereas the above principle asserts that light exhibits both of these characteristics. In this sense, we see that the wave-particle duality commonly associated with quantum mechanics is actually at the core of special relativity as well. (It's not surprising that Einstein was occupied with a paper on light quanta at the same time that he was formulating his ideas about relativity.)

Expressing principles E1 and E2 in the above form enables us to clearly distinguish between the aspects of Einstein's relativistic interpretation that are purely conventional and those that have actual physical content. For example, it is sometimes claimed that the isotropy of the one-way speed of light in special relativity is purely conventional, with no empirical content, since we could just as well select a different system of coordinates in which the speed of light is not isotropic. Furthermore, if we allow the speed of light to be non-isotropic, we could restrict ourselves to systems of coordinates sharing a unique foliation of spacetime into timeslices, making them attractive to people who regard such uniqueness as indispensible (even if the unique foliation must be arbitrarily selected). We discuss the viability of such non-isotropic coordinates in Chapter 4.5. However, the fact that non-isotropic coordinate systems are possible should not obscure the empirical content of Einstein's second principle, which asserts an observable fact, not a mere convention. It is an empirical fact that the propagation of light is invariant with respect to every system of inertial coordinates, given that inertial coordinates are defined such that mechanical inertia is isotropic.

The empirical correspondence between inertial isotropy and lightspeed isotropy can be illustrated by a simple experiment. Three objects, A, B, and C, at rest with respect to each other can be arranged so that one of them is at the midpoint between the other two (the midpoint having been determined using standard measuring rods at rest with respect to those objects). The two outer objects, A and C, are equipped with identical clocks, and the central object, B, is equipped with two identical cannons. Let the two cannons in the center be fired simultaneously in opposite directions toward the two outer objects, and then at a subsequent time let object B emit a flash of light. If the arrivals of the cannonball and light coincide at A, then they also coincide at C, signifying that the propagation of light is isotropic with respect to the same system of coordinates in terms of which mechanical inertia is isotropic, as illustrated in the figure below.

The fact that light emitted from object B propagates isotropically with respect to B's inertial rest frame might seem to suggest that light can be treated as an inertial object within the Galilean framework, just like cannon-balls. However, we also find that if the light is emitted at the same time and place from an object D that is moving with respect to B (as shown in the figure above), the light's speed is still isotropic with respect to B's inertial rest frame. Now, this might seem to suggest that light is a disturbance in a material medium in which the objects A,B,C just happen to be at rest, but this is ruled out by the fact that it applies regardless of the state of (uniform) motion of those objects. Naturally this implies that the flash of light propagates isotropically with respect to the inertial rest coordinates of object D as well. To demonstrate this, we could arrange for two other bodies, denoted by E and F, to be moving at the same speed as D, and located an equal distance from D in opposite directions. Then we could fire two identically constructed cannons (at rest with respect to D) in opposite directions, toward E and F. The results are illustrated below.

The cannons are fired from D when it crosses the x axis, and the cannon-balls strike E and F at the events marked a and b, coincident with the arrival of the light pulse from D. Obviously the time axis for the inertial rest frame coordinates of object D is the worldline of D itself (rather than the original "t" axis shown on the figure). In addition, since inertial coordinates are defined such that mechanical inertia is isotropic, it follows that the cannon-balls fired from identical cannons at rest with D are moving with equal and opposite speeds with respect to D's inertial rest coordinates, and since E and F are at equal distances from D, it also follows that the events a and b are simultaneous with respect to the inertial rest coordinates of D. Hence, not only is the time axis of D's rest frame slanted with respect to B's time axis, the spatial axis of D's rest frame is equally slanted with respect to B's spatial axis.

Several other important conclusions can be deduced from this figure. For example, with respect to the original x,t coordinate system, the speeds of the cannon-balls from D are not given by simply adding (or subtracting) the speed of the cannon-balls with respect to D's rest frame to (or from) the speed of D with respect to the x,t coordinates. Since momentum is explicitly conserved, this implies that the inertia of a body increases with it's velocity (i.e., kinetic energy), as is discussed in more detail in Section 2.3. We should also note that although the speed of light is isotropic with respect to any inertial spacetime coordinates, independent of the motion of the source, it is not correct to say that the light itself is isotropic. The relationship between the frequency (and energy) of the light with respect to the rest frame of the emitting body and the frequency (and energy) of the light with respect to the rest frame of the receiving body does depend on the relative velocity between those two massive bodies (as discussed in Chapter 2.4).

Incidentally, notice that we can rule out the possibility of object B and D dragging the light medium along with them, because they are moving through the same region of space at the same time, and they can't both be dragging the same medium in opposite directions. This is in contrast to the case of (for example) accoustic pressure waves in a material substance, because in that case a recognizable material substance determines the unique isotropic frame, whereas in the case of light we're unable to identify any definite material medium, so it the medium has no definite rest frame.

The first person to discern the true relationship between relatively moving systems of inertial coordinate systems was Lorentz. Not surprisingly, he arrived at this conception in a rather indirect and laborious way, and didn't immediately recognize that the class of coordinate systems which he had discovered (and which he called "local coordinate" systems) were none other than Galileo's inertial coordinate systems. He began with the absolute ether frame coordinates t and x, so every event can be assigned a unique space-time position (t,x) in terms of these absolute coordinates, and then he considers a system moving with the velocity v in the positive x direction. He applied the traditional Galilean transformation to assign a new set of coordinates to every event. Thus an event with ether-frame coordinates t,x is assigned the new coordinates x" = x - vt and t" = t. Then he tentatively proposed an additional transformation that must be applied to x",t" in order to give the "local" coordinates, in terms of which Maxwell's equations apply in their standard form. Lorentz was not entirely clear about the physical significance of these coordinates, but it turns out that all physical phenomena conform to the same isotropic laws of physics when described in terms of these coordinates. (Lorentz's notation made use of the parameter b = 1/g = 1/(1-v²)^1/2 and another constant which he later determines to be 1.) Taking units such that c = 1, his equations for the local coordinates x' and t' in terms of the Galilean coordinates which we're calling x" and t" are

Recall that the traditional Galilean transformation is x" = x - vt and t" = t, so we can make these substitutions to give the complete transformation from the original ether rest frame coordinates x,t to the local coordinates moving with speed v

These effective coordinates enabled Lorentz to explain how two relatively moving observers, each using his own local system of coordinates, both seem to remain at the center of expanding spherical light waves originating at their point of intersection, as illustrated below

The x and x' axes represent the respective spatial coordinates (say, in the east/west direction), and the t and t' axes represent the respective time coordinates. One observer is moving through time along the t axis, and the other has some relative westward velocity as he moves through time along the t' axis. The two observers intersected at the event labeled O, where they each emitted a pulse of light. Those light pulses emanated away from O along the dotted lines. Subsequently the observer moving along the t axis finds himself at C, and according to his measures of space and time the outward going light waves are at E and W at that same instant, which places him at the midpoint between them. On the other hand, the observer moving along t' axis finds himself at point c, and according to his measures of space and time the outward going light waves are at e and w at this instant, which implies that he is at the midpoint between them.

Thus Lorentz discovered that by means of the "fictitious" coordinates x',t' it was possible to conceive of a class of relatively moving coordinate systems with respect to which the speed of light is invariant. However, Lorentz was dissatisfied with the proliferation of ad hoc hypotheses that he had made in order to arrive at this theory. It was in a review of Lorentz's work that Poincare remarked "hypotheses are what we lack least". Lorentz continued to ponder the matter. In 1904 he wrote

In the next section we will review Lorentz's "better result", which consists of a plausibility argument for why physical objects conform to the "local coordinates" that were essentially just determining the transformation under which Maxwell's equations maintain their form. Before we consider Lorentz's attempt at what Einstein later called a "constructive theory" (as opposed to a theory of principle), it's worth noting that Lorentz's transformation had already been derived by Voldemar Voigt in 1887 based simply on the ordinary wave equation. Recall that the wave equation for a time-dependent scalar field f(x,t) in one dimension is

where u is the propagation speed of the wave. This equation was first studied by Jean d'Alembert in the 18th century, and it applies to a wide range of physical phenomena. In fact it seems to represent a fundamental aspect of the relationship between space, time, and motion, transcending any particular application. Traditionally it was considered to be valid only for a coordinate system x,t with respect to which the wave medium (presumed to be an inertial substance) is at rest and has isotropic properties, because if we apply a Galilean transformation to these coordinates, the wave equation is not satisfied with respect to the transformed coordinates. However, Galilean transformations are not the most general possible linear transformations. Voigt considered the question of whether there is any linear transformation that leaves the wave equation unchanged.

The general linear transformation between (x,t) and (X,T) is of the form

for constants A,B,C,D. If we choose units of space and time so that the speed u equals 1, the wave equation in terms of (X,T) is simply �²f/�X² = �²f/�T². To express this equation in terms of the original (x,t) coordinates, recall that the total differential of f can be written in the form

Also, at any constant T, the value of f is purely a function of X, so we can divide through the above equation by dX to give

Taking the partial derivative of this with respect to X then gives

Since partial differentiation is commutative, this can be written as

Substituting the prior expression for �f/dX and carrying out the partial differentiations gives an expression for �²f/�X² in terms of partials of f with respect to x and t. Likewise we can derive an expression for �²f/�T². Substituting into the wave equation gives

This is equivalent to the condition that f(X,T) is a solution of the wave equation with respect to the X,T coordinates. Since the mixed partial generally varies along a path of constant second partial with respect to x or t, it follows that a necessary and sufficient condition for f(x,t) to also be a solution of the wave equation in terms of the x,t coordinates is that the constants A,B,C,D of our linear transformation satisfy the relations

Furthermore, the differential of the space transformation is dx = AdX + BdT, so an increment with dx = 0 satisfies dX/dT = -B/A. This represents the velocity at which the spatial origin of the x,t coordinates is moving relative to the X,T coordinates. We will refer to this velocity as v. We also have the inverse transformation from (X,T) to (x,t):

Proceeding as before, the differential of this space transformation gives dx/dt = B/D for the velocity of the spatial origin of the X,T coordinates with respect to the x,t coordinates, and this must equal -v. Therefore we have B = -Av = -Dv, and so A = D. It follows from the condition imposed by the wave equation that B = C, so both of these equal -Av. Our transformation can then be written in the form

and the inverse transformation is

The determinant of the transformation is A²(1-v²), so to make this equal to 1 we must have A = 1/(1-v²)^1/2. Aside from incidental scale factors, this is how Voigt derived the "Lorentz transformation". He evidently regarded the transformed coordinates x and t as merely a convenient parameterization for purposes of calculation, and didn't attach any greater significance.

Lorentz, too, initially considered his "local coordinates" to be nothing more than an aid to calculation. It's interesting that (unlike Voigt) Lorentz derived the transformation in two separate stages. He first developed the "local time" coordinate, and only years later came to the conclusion (after but independently of Fitzgerald) that a "contraction" of spatial length was also necessary in order to account for the absence of second-order effects in Michelson's experiment. This is sometimes said to have been an "ad hoc" assumption on the part of Lorentz, although he was able to give a plausibility argument for the contraction based on what he called the Molecular Force Hypothesis and his theorem of Corresponding States, as discussed in the next section.